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In this lecture by Frederic schuller, he introduces the the covariant derivative axiomatically:

A connection $\nabla$ on a smooth manifold $(M,O,A)$ is a map that takes a pair consisting of a vector (field) $X$ and a $(p,q)$ - Tensor field $T$ and sends them to a $(p,q)$ tensor field $\nabla_X T$ following these axioms:

  1. For a scalar function $\nabla_X f = Xf$ where $Xf$ denotes the directioncal derivative of $f$ in the direction of $X$

  2. $\nabla_X (T+S) =\nabla_X T + \nabla_X S$

  3. $\nabla_X T( \omega,Y) = (\nabla_X T)(\omega,Y) + T(\nabla_X \omega, Y) + T(\omega, \nabla_X Y)$ (Leibniz law)

  4. $\nabla_{fX+Z} T = f \nabla_X T+ \nabla_Z T$

Later in the lecture prof. Schuller remarks that for the of operators satisfying the above axioms, the only freedom we have left in saying how it behaves is the christoffel symbols, i.e: how to the connection acts on the basis vectors of the chart (See 34:22). This action is specified by the Christoffel symbols.

The motif in the above is the above, to my understanding, is that when we specify the Christoffel symbols we say exactly how the surface sits in space. (Related)

A yet another approach is taken by prof.Pavel Grinfeld in his lecture series for Tensor analysis, in his engineering style of explanation, he says that the Christoffel's is defined as the derivative of the covariant basis with respect to coordinates (1:07 here)

The Christoffel comes as:

$$ \frac{\partial \vec{Z_i} }{\partial Z_i} = \Gamma_{kj}^i \vec{Z_i} \tag{1}$$

Earlier in the lecture series, it was defined that the covariant basis is given by derivative of the position vector parameterized with coordinates (4:52, he takes about an example of this)i.e:

$$ \vec{Z_i} =\frac{\partial R(Z_1,Z_2,Z_3) }{\partial Z^i} \tag{2}$$

When one looks at (2) then (1), it seems that Christoffel would naturally come when one tries to take derivative of a parameterized position vector. But, in prof. Schuller's lecture, it seems that one must to define what a connection is and only when one defines the action of connection do Christoffel's get defined.

Could someone explain how these two different approaches fit together?

Clemens Bartholdy
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  • $(1)$ looks to be a typo. It should read something like$$\frac{\partial Z^i}{\partial Z^j}=\Gamma^i{}_{jk}Z^k$$Also, the partial derivative in $(1)$ and $(2)$ only makes sense if the manifold $M$ is embedded in $\mathbb{R}^N$, whereas the axiomatic approach is usually used for abstract manifolds. – Kajelad Mar 28 '22 at 01:56
  • How so , aren't covariant derivative controlled by Christo symbols? @Kajelad – Clemens Bartholdy Mar 28 '22 at 02:37
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    I'm not sure what you mean. On an abstract manifold, one can either define covariant differentiation in terms of Christoffel symbols, or define it axiomatically and derive the behavior of Christoffel symbols from the axioms. – Kajelad Mar 28 '22 at 03:45
  • If you look at my recent edit to my answer (the one where you linked this question) I think you'll find the connection between these two approaches. Also, as mentioned above, this approach only makes sense in the very special case of an embedded submanifold, with the induced Levi-Civita connection. – peek-a-boo Mar 28 '22 at 11:31
  • First of all, it should be $$\mathbf{\vec{Z}}^i\cdot \frac{\partial \mathbf{\vec{Z}}j}{\partial Z^k}=\Gamma^i{jk}$$ or $$\mathbf{\vec{Z}}j\cdot \frac{\partial \mathbf{\vec{Z}}^i}{\partial Z^k}=-\Gamma^i{jk}$$ with $$\mathbf{\vec{Z}}^i \cdot \mathbf{\vec{Z}}_j=\delta^i_j$$. Also, as I have seen this notation before in questions here. Why are you using $$Z$$ for coordinates, basis vectors and everything? – ContraKinta Mar 29 '22 at 03:37
  • Also note that this "definition" assumes a scalar product so you have a metric. $$$$ On one hand, the existence of a metric ensures the existence of an associated connection together with a corresponding theory of curvature. On the other hand, the existence of a connection alone does not in general imply the existence of a metric. One of the most important insights when you learn about this stuff is that the mere existence of an affine connection is sufficient for the purposes of a self contained theory of tensors. – ContraKinta Mar 29 '22 at 04:03
  • That notation, it's the sasme as one used in Pavel Grinfeld's book and lectures. The coordinates are $(Z_1,Z_2,...)$ . If you could write your point as an answer , I'll accept it @ContraKinta – Clemens Bartholdy Mar 30 '22 at 20:15

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